NAME

Algorithm::Munkres - Perl extension for Munkres' solution to 
classical Assignment problem for square and rectangular matrices 
This module extends the solution of Assignment problem for square
matrices to rectangular matrices by padding zeros. Thus a rectangular 
matrix is converted to square matrix by padding necessary zeros.

SYNOPSIS

use Algorithm::Munkres;

    @mat = (
	 [2, 4, 7, 9],
	 [3, 9, 5, 1],
	 [8, 2, 9, 7],
	 );

assign(\@mat,\@out_mat);

Then the @out_mat array will have the output as: (0,3,1,2),
where 
0th element indicates that 0th row is assigned 0th column i.e value=2
1st element indicates that 1st row is assigned 3rd column i.e.value=1
2nd element indicates that 2nd row is assigned 1st column.i.e.value=2
3rd element indicates that 3rd row is assigned 2nd column.i.e.value=0

DESCRIPTION

    Assignment Problem: Given N jobs, N workers and the time taken by 
    each worker to complete a job then how should the assignment of a 
    Worker to a Job be done, so as to minimize the time taken. 

	Thus if we have 3 jobs p,q,r and 3 workers x,y,z such that:
	    x  y  z		
	 p  2  4  7
	 q  3  9  5
	 r  8  2  9
        
        where the cell values of the above matrix give the time required
        for the worker(given by column name) to complete the job(given by 
        the row name) 
    
	then possible solutions are:	
		 	 Total
	 1. 2, 9, 9       20
	 2. 2, 2, 5        9
	 3. 3, 4, 9       16
	 4. 3, 2, 7       12
	 5. 8, 9, 7       24
	 6. 8, 4, 5       17

    Thus (2) is the optimal solution for the above problem.
    This kind of brute-force approach of solving Assignment problem 
    quickly becomes slow and bulky as N grows, because the number of 
    possible solution are N! and thus the task is to evaluate each 
    and then find the optimal solution.(If N=10, number of possible
    solutions: 3628800 !)
    Munkres' gives us a solution to this problem, which is implemented 
    in this module.

    This module also solves Assignment problem for rectangular matrices 
    (M x N) by converting them to square matrices by padding zeros. ex:
    If input matrix is:
	 [2, 4, 7, 9],
	 [3, 9, 5, 1],
	 [8, 2, 9, 7]
    i.e 3 x 4 then we will convert it to 4 x 4 and the modified input 
    matrix will be:
	 [2, 4, 7, 9],
	 [3, 9, 5, 1],
	 [8, 2, 9, 7],
 	 [0, 0, 0, 0]

EXPORT

"assign" function by default.

INPUT

The input matrix should be in a two dimensional array(array of 
array) and the 'assign' subroutine expects a reference to this 
array and not the complete array. 
eg:assign(\@inp_mat, \@out_mat);
The second argument to the assign subroutine is the reference 
to the output array.

OUTPUT

The assign subroutine expects references to two arrays as its 
input paramenters. The second parameter is the reference to the
output array. This array is populated by assign subroutine. This 
array is single dimensional Nx1 matrix.
For above example the output array returned will be:
 (0,
 2,
 1)

where 
0th element indicates that 0th row is assigned 0th column i.e value=2
1st element indicates that 1st row is assigned 2nd column i.e.value=5
2nd element indicates that 2nd row is assigned 1st column.i.e.value=2

SEE ALSO

1. http://216.249.163.93/bob.pilgrim/445/munkres.html

2. Munkres, J. Algorithms for the assignment and transportation 
   Problems. J. Siam 5 (Mar. 1957), 32-38

3. François Bourgeois and Jean-Claude Lassalle. 1971.
   An extension of the Munkres algorithm for the assignment 
   problem to rectangular matrices.
   Communication ACM, 14(12):802-804

AUTHOR

    Anagha Kulkarni, University of Minnesota Duluth
    kulka020 <at> d.umn.edu
	
    Ted Pedersen, University of Minnesota Duluth
    tpederse <at> d.umn.edu

COPYRIGHT AND LICENSE

Copyright (C) 2005-2006, Ted Pedersen and Anagha Kulkarni

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.